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Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers

Abstract : We are interested in this thesis by the analysis of scattering and inverse scattering problems for locally perturbed periodic infinite layers at a fixed frequency. This problem has connexions with non destructive testings of periodic media like photonics structures, optical fibers, gratings, etc. We first analyze the forward scattering problem and establish some conditions under which there exist no guided modes. This type of conditions is important as it shows that measurements can be done on a layer above the structure without loosing substantial informations in the propagative part of the wave. We then propose a numerical method that solves the direct scattering problem based on Floquet-Bloch transform in the periodicity directions of the background media. We discretize the problem uniformly in the Floquet-Bloch variable and use a spectral method in the space variable. The discretization in space exploits a volumetric reformulation of the problem in a cell (Lippmann-Schwinger integral equation) and a periodization of the kernel in the direction orthogonal to the periodicity. The latter allows the use of FFT techniques to speed up Matrix-Vector product in an iterative to solve the linear system. One ends up with a system of coupled integral equations that can be solved using a Jacobi decomposition. The convergence analysis is done for the case with absorption and numerical validating results are conducted in 2D. For the inverse problem we extend the use of three sampling methods to solve the problem of retrieving the defect from the knowledge of mutistatic data associated with incident near field plane waves. We analyze these methods for the semi-discretized problem in the Floquet-Bloch variable. We then propose a new method capable of retrieving directly the defect without knowing either the background material properties nor the defect properties. This so-called differential-imaging functional that we propose is based on the analysis of sampling methods for a single Floquet-Bloch mode and the relation with solutions toso-called interior transmission problems. The theoretical investigations are corroborated with numerical experiments on synthetic data. Our analysis is done first for the scalar wave equation where the contrast is the lower order term of the Helmholtz operator. We then sketch the extension to the cases where the contrast is also present in the main operator. We complement our thesis with two results on the analysis of the scattering problem for periodic materials with negative indices. Weestablish the well posedness of the problem in 2D in the case of a contrast equals -1. We also show the Fredholm properties of the volume potential formulation of the problem using the T-coercivity approach in the case of a contrast different from -1.
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Submitted on : Sunday, April 16, 2017 - 7:51:09 PM
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Thi Phong Nguyen. Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers. Analysis of PDEs [math.AP]. Université Paris Saclay (COmUE), 2017. English. ⟨NNT : 2017SACLX004⟩. ⟨tel-01475424v2⟩

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